# Fractional decompositions and the smallest-eigenvalue separation

## TL;DR

This paper presents a novel approach using fractional graph decompositions to bound the eigenvalue separation in non-bipartite regular graphs, providing a concise proof of a recent eigenvalue result.

## Contribution

Introduces a new fractional decomposition method to bound eigenvalue separation, simplifying proofs of existing theorems in spectral graph theory.

## Key findings

- Bound on eigenvalue separation established
- Simplified proof of eigenvalue bounds for distance-regular graphs
- Enhanced understanding of spectral properties of non-bipartite graphs

## Abstract

A new method is introduced for bounding the separation between the value of $-k$ and the smallest eigenvalue of a non-bipartite $k$-regular graph. The method is based on fractional decompositions of graphs. As a consequence we obtain a very short proof of a generalization and strengthening of a recent result of Qiao, Jing, and Koolen [Non-bipartite distance-regular graphs with a small smallest eigenvalue, Electronic J. Combin. 26(2) (2019), P2.41] about the smallest eigenvalue of non-bipartite distance-regular graphs.